Let $\{\phi_1,\phi_2,.......\}$ be a row orthogornal polynomials which are not zero. With $\phi_i \in P_i$ and inner product $<.,.>$ on $C\left([a,b]\right)$.
show that $\{\phi_1,................,\phi_n\}$ is an basis of $P_n$ and prove that $\phi_n$ is exactly of degree n
i've tried to prove it by proving the lineair independance property and the spanning property.
The lineair independance property wasn't that hard but i dont see how to prove the spanning property. Can i get a hint/tip on that.
And with gram-shmidt we see that: $\phi_n = x^{n}-\alpha_0\phi_0-....-\alpha_{n-1}\phi_{n-1}$ so isn't it trivial to see that $\phi_n$ is a n'th degree polynomial?